MatheInatics for QuantUIn Mechanics An Introductory Sur"e.y of Operators, Eigen"alues, and Linear Vector Spaces John David Jackson University of Illinois w. New York 1962 www.com MATHEMATICS FOR QUANTUM MECHANICS An Introductory Survey Copyright@ 1962 by W. All rights reserved. Library of Congress Catalog Card Number: 62-17526 Manufactured in the United States of America The manuscript was received April 1, 1962, and published July 20, 1962.com Editor's Foreword Everyone concerned with the teaching of physics at the advanced undergraduate or graduate level is aware of the continuing need for a modernization and reorganization of the basic course material.
Despite the existence today of many good textbooks in these areas, there is always an appreciable time-lag in the incorporation of new view-points and techniques which result from the most recent de- velopments in physics research. Typically these changes in con- cepts and material take place first in the personal lecture notes of some of those who teach graduate courses. Eventually, printed notes may appear, and some fraction of such notes evolve into textbooks or monographs. But much of this fresh material remains available only to a very limited audience, to the detriment of all.
Our series aims at filling this gap in the literature of physics by presenting occasional volumes with a contemporary approach to the classical topics of physics at the advanced undergraduate and graduate level. Clarity and soundness of treatment will, we hope, mark these volumes, as well as the freshness of the approach. Another area in which the series hopes to make a contribution is by presenting useful supplementing material of well-defined scope. This may take the form of a survey of relevant mathematical prin- ciples, or a collection of reprints of basic papers in a field.
Here the aim is to provide the instructor with added flexibility through the use of supplements at relatively low cost. The scope of both the lecture notes and supplements is somewhat different from the "Frontiers in Physics" series. In spite of wide variations from institution to institution as to what comprises the basic graduate course program, there is a widely accepted group of "bread and butter" courses that deal with the classic topics In physh::s. ThE)SC include: Mathematical methods of physics, www.com vi EDITORS' FOREWORD electromagnetic theory, advanced dynamics, quantum mechanics, statistical mechanics, and frequently nuclear physics and/or solid state physics.
It is chiefly these areas that will be covered by the present series. The listing is perhaps best described as including all advanced undergraduate and graduate courses which are at a level below seminar courses dealing entirely with current research topics. The publishing format for the series is in keeping with its inten- tions. Photo-offset printing is used throughout, and the books are paperbound in order to speed publication and reduce costs.
It is hoped that books will thereby be within the financial reach of grad- uate students in this country and abroad. Finally, because the series represents something of an experi- ment on the part of the editors and the publisher, suggestings from interested readers as to format, contributors, and contributions will be most welcome. DAVID JACKSON DAVID PINES www.com Preface The purpose of these notes is to present concisely the mathemat- ical methods of quantum mechanics in a form which emphasizes the unity of the different techniques. Since the methods are applicable to the description of many physical systems outside the domain of quantum theory, the material may be useful in other areas.
But the orientation is toward the graduate or advanced undergraduate student beginning a serious study of quantum mechanics. The notes were de- veloped as a supplement for the first-year graduate course in quan- tum mechanics at the University of Illinois. At all but a few graduate schools in physics, the entering students come with a variety of mathematical backgrounds, ranging from ig- norance of Fourier series and partial differential equations, on the one hand, to familiarity with group theory and Banach spaces, on the other. The teaching of quantum mechanics to such a heterogeneous group presents problems.
It was in an attempt to solve some of these pr.oblems that the present little volume came into being. My aim was to assure that everyone had a certain level of knowledge in those areas of mathematics that bear most directly on quantum mechanics. The level is not high, to be sure, but it is adequate for the beginning student. When teaching quantum mechanics, I per- sonally spend five or six weeks at the start in covering the material presented here.
Then I feel free to discuss the physical subject with whatever formalism is most appropriate for the topic at hand. But others may wish to discuss the relevant mathematics as the need. arises, or even assume that the student can learn it outside the lec- ture room. Whatever the attitude, I hope that these notes will serve both teacher and student by bringing together in compact form tpe essential mathematical background for quantum mechanics.
JACKSON April 15, 1962 vi i www.com Contents Editors' Foreword v Preface vii 1 Introductory Remarks 1 References 2 2 Eigenvalue Problems in Classical Physics 4 2-1 Vibrating string 4 2-2 Vibrating circular membrane 6 2- 3 Small oscillations of a mechanical system 10 2- 4 Rotation of axes and orthogonal transformations 15 2- 5 Euler's theorem and principal-axes transformations as eigenvalue problems 18 3 Orthogonal Functions and Expansions )2 3-1 Fourier series 22 3-2 Expansion in orthonormal functions 25 3- 3 Dirac delta function and closure relation 28 3- 4 Bessel functions as an orthonormal set on the interval (0,1) 31 3- 5 Schmidt orthogonalization method 33 3-6 Legendre polynomials 35 www.com x CONTENTS 3-7 Other orthogonal polynomials 36 3- 8 Fourier integrals 37 4 Sturm-Liouville Theory and Linear Operators on Functions 41 4-1 Sturm-Liouville eigenvalue problem 41 4-2 Linear operators on functions 43 4- 3 Eigenvalue problem for a linear Hermitean operator 45 4-4 Further properties of operators 46 5 Linear Vector Spaces 48 5-1 State vectors and representatives 48 5-2 Complex vectors in a n-dimensional Euclidean space 49 5-3 Basis and base vectors 51 5-4 Change of basis 53 5- 5 Linear operators and their matrix representation 56 5-6 Further definitions and properties of linear operators 59 5-7 Unitary operators and equations of motion 63 5-8 Eigenvectors, eigenvalues, and spectral representation 66 5-9 Determination of eigenvalues and eigenvectors 68 5-10 Transition to Hilbert space; Dirac notation 72 5-11 State vectors and wave functions 75 Appendix A: Bessel (Cylinder) Functions 79 Appendix B: Legendre Functions and Spherical Harmonics 88 www.com Mathematics for Quantum Mechanics www.com 1 Introductory Remarks The purpose of these notes is to set forth the essentials of the mathematics of quantum mechanics with only enough mathematical rigor to avoid mistakes in the physical applications. In various parts of quantum theory it is appropriate to use math- ematical methods that at first sight are quite different and uncon- nected. Thus in dealing with potential barriers or the hydrogen atom, the techniques of ordinary or partial differential equations in coordi- nate space are employed, whereas for a problem such as the har- monic oscillator, the use of abstract linear operator methods leads to an elegant solution. The chief aims of the present discussion are to show the underlying unity of all the methods and to build up enough familiarity with each of them so that in the subsequent treatment of quantum mechanics as a subject of physics the best method can be applied to each problem without apology and without the need to ex- plain new mathematics.
Quantum theory was originally developed with two different math- ematical techniques- Schrodinger wave mechanics (differential equa- tions) and Heisenberg matrix mechanics. The equivalence of the two approaches was soon demonstrated, and the mathematical methods were generalized by Dirac, who showed that the techniques of Heisen- berg and Schrodinger were special representations of the formalism of linear operators in an abstract vector space. The concept of discreteness is central in quantum theory. Physi- cally measurable quantities (called "observables") are often found to take on only certain definite values, independent of external con- ditions (e., preparation of light source, detailed design of deflecting magnet, etc.
Important examples of discrete observables are en- ergy (Ritz combination principle and Rydberg formula, Franck-Hertz oxpertnlent) and anp;ular momenlum (Stern-Gerlach experiment).com 2 MATHEMATICS FOR QUANTUM MECHANICS mathematical language the discrete, allowed values of an observable are called eigenvalues (sometimes called characteristic or proper values). The physicist is often interested in predicting and corre- lating the eigenvalues for a given physical system. Provided he has an appropriate mathematical description of the physical system, he wants to solve "the eigenvalue problem." Thus the mathematical eigenvalue problem is an important aspect of quantum mechanics. This problem can be phrased in terms of differential equations, in terms of matrices, or in terms of linear vector spaces.
We shall consider the various techniques and explore their essential unity below. Not all of quantum mechanics concerns itself with discrete eigenvalues, of course. Hence some of the mathematical discussion will not bear directly on the eigenvalue problem. Furthermore, a number of topics, such as perturbation theory and variational meth- ods, will be omitted from these notes, to be dealt with separately.
References Since only the bare essentials will be presented in these notes, the student will wish to consult more complete treatments. Some useful references are the following: R. Hilbert, "Methods of Mathematical Physics," Vol. I, Interscience, New York, 1953.
Chapter 2 on orthogonal expansions; Chap. 5 on eigenvalue problems; Chap. 7 on special functions. Friedman, "Principles and Techniques of Applied Mathematics," Wiley, New York, 1956.
A very useful, if formal, treatment. Tralli, "Some Mathematical Methods of Physics," McGraw-Hill, New York, 1960. Halmos, "Finite Dimensional Vector Spaces," Princeton Uni- versity Press, Princeton, N., 1942; "Introduction to Hilbert Space," Chelsea, New York, 1~51. These books present a rigor- ous mathematical discussion of vector spaces.
Hildebrand, "Methods of Applied Mathematics," Prentice-Hall, Englewood Cliffs, N. The first 100 pages deal with mat- rices and vector spaces. Murphy, "Mathematics of Physics and Chem- istry," 2nd ed., Van Nostrand, Princeton, N. Chapters 2, 3, 7, 8, 10, and 11 have particular relevance.
Feshbach, "Methods of Theoretical Physics," 2 vols., McGraw-Hill, New York, 1953. Very complete, with valu- able appendices at the end of each chapter. Rojansky, "Introductory Quantum Mechanics," Prentice-Hall, Englewood Cliffs, N. Chapter IX has a useful review of matrices.
Parts of Chaps. X and XI are also relevant.com INTRODUCTORY REMARKS 3 H. Sagan, "Boundary and Eigenvalue Problems in Mathematical Physics," Wiley, New York, 1961. Sommerfeld, "Partial Differential Equations," Academic, New York, 1949.
The emphasis is on orthonormal expansions, special functions, eigenfunctions, and eigenvalues. Special mention must be made of one extremely useful reference on special functions; w. Oberhettinger, "Formulas and Theorems for the Special Functions of Mathematical Physics," Chelsea, New York, 1949. This book will be referred to often and will be denoted as "MO." A much more elaborate collection of information on special func- tions and various transforms is contained in the Bateman volumes, Bateman Manuscript Project, "Higher Transcendental Functions," 3 vols.), McGraw-Hill, New York, 1953.
Bateman Manuscript Project, "Tables of Integral Transforms," 2 vols.), McGraw-Hill, New York, 1954. Note, however, that MO has quite useful tables of Fourier and Laplace transforms. When it comes to tables of integrals and numerical values of the elementary functions, personal preference takes over. Useful, in- expensive references include the following: H.
Dwight, "Tables of Integrals and Other Mathematical Data," Macmillan, New York. Foster, "A Short Table of Integrals," 4th ed.k of Chemistry and Physics, "Mathematical Tables," McGraw-Hill, New York. Emde, "Tables of Functions" (paperback), Dover, New York, 1945. Tabulation and graphs of special functions.
As a final introductory remark let me say that it is assumed that the student has some familiarity with ordinary differential equations, the method of separation of variables for partial differential equa- • tions, the elements of Fourier series, the elementary properties of matrices, and the notions of vectors in three dimensions, rotations, etc.